The cylindrical pressure vessel above has closed ends and contains a fluid at
gauge pressure P as shown below. The outer diameter is D and the wall
thickness.
Thin Walled Pressure vessels
The cylindrical pressure vessel above has closed ends and contains a fluid at gauge pressure P as shown below. The outer diameter is D and the wall thickness is t. The term ‘thin-wall’ may be taken to mean that D/t > 10.
Fluid at pressure P
t
D
σc P σc
L
If we section the cylinder, of length L and its contents across its diameter as seen above, we see that we must have equilibrium of the forces due to the internal pressure P and the circumferential stress σc in the wall. PLD = 2σcLt or,
σc = Pr/t is the circumferential stress in the wall.
Note that have assumed that the stress is uniform across the thickness and that we have ignored the fact that the pressure acts on an area defined by the inner diameter. These are only acceptable if D/t > 10.
σa σa
P
If the cylinder has closed ends, the axial stress σa in the wall I found in a similar way by considering a transverse section as shown above. Equilibrium of forces gives: Pπr2 = σaπDt and thus the axial stress σa = Pr/2t The same assumptions apply. Note that σc and σa are principal stresses and remember that the third principal stress σ3 = 0. The maximum shear stress is thus τmax = | σ1 – σ3 |/2 = pr/2t A thin-wall spherical vessel can be analyzed in the same way and it is easily seen that σc and σa are equal and equal to pr/2t. Thus the principal stresses σ1 and σ2 are equal and σ3 =0. The maximum shear stress is τmax = | σ1 – σ3 |/2 = pr/4t
